Let $f(x, y) = x^3 y^{- 2} e^{- x^2 / y}$ if $y > 0$, $f (x, y) = 0$ if $y \leq 0$. Show that $f(x, y)$ is of class $C^1$ as a function of $x$ for each fixed $y$ and as a function of $y$ for each fixed $x$, but that $f$ is unbounded in any neighborhood of the origin.
This is my attempt to showing that $f$ is of class $C^{1}$ of $x$ for each fixed $y$, but I do not know how to prove $f$ is $C^{1}$ of $y$ for each fixed $x$, or why $f$ is unbounded near the origin:
Substituting $y = a$ with $a \in \mathbb{R}$. If $a > 0$, then $f(x , a) = x^3 a^{- 2} e^{- x^2 / a}$ is composed of elementary functions and is therefore $C^{1}$ on $(0, \infty)$. Otherwise if $a \leq 0$, then $f(x, a) = 0$ is clearly of class $C^1$. Substituting $x = a$ with $a \in \mathbb{R}$, then $f(a, y) = a^3 y^{- 2} e^{- a^2 / y}$ is composed of elementary functions. So $f(x, y)$ is of class $C^1$ as a function of $x$ for each fixed $y$ and as a function of $y$ for each fixed $x$.
Hint for unboundedness: Consider $f$ along the parabola $y=x^2.$